Integrating 1/u with the Natural Log

Why the power rule breaks at $n = -1$ and why $\ln|u|$ fills the gap

Recognizing $\int \frac{1}{u}\,du$ vs. a substitution that produces a power of $\ln$

Why $(\ln x)^{3/2} \neq \frac{3}{2}\ln x$ — and choosing $u$ wisely